In the mean time, we do the same with our setup. Now where our infinite plane was, we put a sheet of paper.

A sheet of paper equals to a very rectangular thin cuboid. Should I imagine a very thin rectangular cuboid centered around the plane ?

No. I find it strange that once people get into 4D, they get super-vision that allows them to see "atoms" and such, lol. I mean, the resolution should stay the same. It's the same sheet of paper viewed in either 3D or 4D. It does not look like a thin cuboid in 3D. It's a sheet of paper, forgodssake.

My latest reflection on 4dspace's misconception is that he is probably confusing hyperspace (in the sense of parallel universes, which remain 3d), with hyperspace (ie 4d). Most of his comments only make sense if you suppose that all viewing is done in the same 3space that the image rests.

In a projection, the points on the image take the colour of the first non-clear point of the ray extending from the plane. The lines may be variously be orthogonal to the plane or converge on a point on the other side of the plane (the image of the 'eye' or POV).

In four dimensions, such rays cross the 3space exactly once. The image of the surface of the 4thing is the entirity of the 3image, not just a viewable surface. When we look at it from the side in 3space, we are much the same as looking at a game of chess at the level of the board, and trying to peice together the depth of location of the peices. This is what is being implied with the projections of 2d onto 1d.

However, it is not the image that is the real thing, but a simple projection. As i mentioned elsewhere, rotating the object in four dimensions, will make the projection different. Rotating the image in three dimensions, is like people who turn maps around, to make the upward direction of the map agree with the motion of the car. It does not change the projected image, but just allows for some sense to be made of the image.

The dream you dream alone is only a dream
the dream we dream together is reality.

Hey Ovo! Where did you go? Did you router get fried too? Just as I got you ready and warm for the kill, you disappear on me. It's not fair

Ovo wrote:

4Dspace wrote:No. The fact that only one side of a plane is visible to a POV in a Euclidean space is a basic fact that you can check yourself.

No, it's not a self-evident fact, it's a wrong fact for me and the other persons in this discussion, and I've spent a lot of time checking the basics of Euclidean geometry and couldn't find a single reference to this. Planes don't have sides with different things on them.

So you've been denying that planes have directions or chirality. Here is the ref from google books, it's Geometry of four dimensions By Henry Parker Manning, 1914, includes Schlafli's work that was published 3 years prior. Here is the relevant quotes from it:

We cannot speak of the right and left sides of a line except as we associate the line with some plane in which it lies; but direction along a line is a property of the line itself and is independent of any plane of space that contains it.

Likewise, we cannot speak of one and the other side of a plane except as we associate the plane with some hyperplane in which it lies; but "order of the plane", or direction of rotation, is independent of any hyperplane.

So, since you cannot deal with a concept of direction of a sheet of paper shown in different colors, because for you it immediately implies a "thin cuboid", I came up with the solution to this problem: Instead of a sheet of paper we have a plane figure proper. And, to show its direction or "order", which is relevant to the discussion at hand, we have a bunch of 2Ders living on this plane (finally! mythical creatures are good for something -- you have to agree that this is in line with the rest of the discussions on this board). Like a line of ants, the 2Drs keep walking in a circle non-stop, in such a way that you can clearly say in which direction (from your POV) they are walking, clockwise or counterclockwise. So, where the sheet of paper had "yes" written, we have 2Ders walk clockwise and where the sheet had "no", the 2Ders walk counterclockwise.

Here. Now you have no excuses. Which way do you see them walk from a POV in a positive quadrant of the axes?

When one stands on the left of a river, it flows one way. When one stands on the right of a river, it flows the other way.

The diverse POV does not imply multiple sides of lines, planes etc, even when chiral objects exist. On the other hand, a chiral object can be reversed in a subspace specifically because it is not solid.

The dream you dream alone is only a dream
the dream we dream together is reality.

I am one and a half years late in joining this forum, and I might not have read the entire page tonight, so please excuse me if the same thing has already been posted. And please excuse me, if any of you have changed your opinions on the matter since this was posted, since my reply is based on your posts one and a half years ago. I have commented on a few quotes of 4Dspace and wendy.

4Dspace wrote:And so, returning to the "problem", i.e. a well-defined cube made of 6 square planes (the exact definition came about not from the start but in the process of discussion in that thread), the task was to see this cube, colored red outside and green inside, in 4D. Of course, in 3D, we cannot see that it is colored green inside. But, having moved our cube into 4D, we can easily see inside it and thus can tell that its 6 planes are colored green inside and red outside.

Outside to what, one may ask. Why, the answer is the same as in 3D: outside to the inner 3d-space this cube occupies. It is still there. It is not 0 as in 3D analogy.

Given the fact that all 6 faces of the cube are visible in their entirety from most positions in 4D, what color they appear to a fixed POV in 4D?

And here, I believe from what you posted above, you would say that some faces will be seen green (the counterclockwise aspect of a writing on its face), and some, red. Agree?

First of all, we don't "move the cube into 4D" (This might be a misunderstanding of terms). We could see the interior as well as the exterior of the cube only with a 3D array of retina cells at the outermost layer. And from such a POV, we would not see some faces as green and the rest as red. First, I would like to emphasize that, in the hollow cube you mention, the fact that the outer side of a face can be coloured differently from the inner side shows that there is a small distance between the two sides, i.e. they are not on the same plane and are not the same square. And I have to use analogy to present my opinion on this. I have given an image, which is of a square, under similar conditions as that of your cube. (My apologies if the image is not successfully posted.)

[img]ikariasquare.bmp[/img]

Suppose there was a Flatlander on the same plane as this square. The Flatlander would see only the outer red side of the square's perimeter. And the faces of the cube as seen from the 4D POV would be similar to the way we see the edges of this square. We cannot say that some sides are red and the rest are green. Similarly, we cannot say that some faces of your cube are red while the others are green.

4Dspace wrote:And so, how do you see the 6 faces of such a cube in 4D? I say, half are seen red and half, green. That's 3 and 3. The 3 'near' ones are red and 3 'far' ones are green.

Now, what's near and far? A hyperplane, which is a 3D object defined by 3 bounding 2d planes orthogonal to each other, separates 4D into 2 halves. It's a wall you cannot penetrate. If the wall is not infinite (as the case is with our cube == it's a bounded 3d subspace in 4D), then you can walk around it. But it's still a "wall" in the sense that you cannot walk through it, and, unless it is transparent, you can't see its other side. You POV determines which side is seen. One at a time.

So, a cube in 4D is bound by "2 parallel hyperplanes" (both parallel to each other and perpendicular to the POV). These two hyperplanes carve out a 3D subspace out of the 4D space they are in. If here we are to use an analogy with 3D, this would be equivalent to two 2d-planes sandwiching a section of 2D space between. Seems redundant, since the 2D space has neither direction nor length in the 3rd dimension (==from 3D POV). Yet, in this analogy that's how it is, a 0. And so, going out of the analogy into what is, we realize that 0 (which is the total volume a 2D plane occupies in 3D) is not equal to x cubed in 4D. Here we make a leap from 2D seen from 3D, to 3D as seen from 4D. There is no correspondence in this step. Here the analogy is misleading: A gazillion of 2d-planes will never amount to a cube in 3D. Simply cause 0 times a gazillion is still 0. But, funny enough (!) eight 3d cubes in 4D do in fact make up a tesseract, a bona fide, real, 4d-object.

I agree that "A gazillion of 2d-planes will never amount to a cube in 3D. Simply cause 0 times a gazillion is still 0." But if it is zero times infinite, then it can amount to any finite value and not zero. This concept should be familiar to anybody who knows calculus. This means that a cube is made of infinite squares stacked together. And as for your statement, "eight 3d cubes in 4D do in fact make up a tesseract, a bona fide, real, 4d-object", it is just as sensible as saying that six 2d squares in 3d do in fact make up a cube, a bonafide, real, 3d object. In other words, the six squares just make up a boundary of the cube, not the entire thing. Similarly, eight cubes just make up the boundary of a tesseract and it has another interior which is 4D and each point inside it can be represented by 4 Cartesian coordinates. You might feel that I am going to the very basic concepts, but I felt that it was necessary from what I understood from your post. It is obvious that you cannot visualise the fourth dimension (neither can I, though the concepts are familiar to me and I find no fault with any of the statements that you have contradicted).

wendy wrote:This view comes from looking at a squashed polytope too. No, it makes a hollow box or the surface. You need to add bulk or substance to it to make a real 4d object, just as you have to fill the six squares to make a cube. Otherwise, it's a drawing on a peice of paper.

I agree with this statement by wendy. Eight cubes connected by their surfaces make a hollow tesseract, and a solid tesseract is made when infinite cubes are stacked together in the "Upsilon"/"Delta" directions.

4Dspace wrote:wendy, this is the last time I answer your posts, since I do not see a point talking to you, a self-appointed expert in seeing higher dimensions, who is incompetent at visualizing a simple cube in 4D. This simple exercise revealed the truth. You claim that you "see up to 8D". Of course, no-one can get into your head and see what you see, but from what you posted above it appears that you are simply deluding yourself.

It is difficult for me to believe that a 3D human can see upto 8D. But where is the citation by 4Dspace which claims that wendy can see eight dimensions? Is this an exaggeration?

Last edited by Prashantkrishnan on Mon Jan 13, 2014 8:16 pm, edited 1 time in total.

People may consider as God the beings of finite higher dimensions,though in truth, God has infinite dimensions

I think I got the misconception in this discution: let's start with a flatlander who has a square and a 2-dimensional pencil. this flatlander could write something on the sides of the square (in morse code or something). now let's say you'd pick this square up. Would you be able to see what is written on the sides? I think there are two answers to this:1. the "non-mathematical" view: No, the writing can't be seen, as it is written on the sides of the squares. Intuitively, you won't be able to see, as you also can't look at the side of a piece of paper to see if there's anything written on it.

2. the "mathematical" view: Yes, the sides of the square are made of points, and by writing on the sides, these points get another color. Now the viewer can look at the edge and see that some points have a different colour.

There's nothing wrong with the second view, and this view is what I think quickfur and wendy use.

But really, I thought this forum was made for discussions rather than arguments.student91

How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.-Stern/Multatuli/Eduard Douwes Dekker

Discussion, i am afraid, includes disagreements. That's the nature of the beast. How can you come to be right, if ye never make a mistake?

Yes, i have been to many strange places, and seen many strange things. The nature of six and seven and eight dimensions are revealed through things one might feel, more than see, but one feels it with one's eyes, one's mind.

The dream you dream alone is only a dream
the dream we dream together is reality.

Visualizing 8-D isn't that tough. No doubt it takes some time, along with developing little tricks that you come to learn, usually on your own. It's a tougher thing to teach entirely, rather than the self-taught method, once the basics have been addressed. Of course, one doesn't dive into 8-D right away. Four and 5-D have to become elementary in understanding, then the higher dimensions are simply more operations to a 5-D base shape.

In response to all of the above posts ( and I'm afraid to get into it at this point) , there is a difference between solid vs translucent objects. That seems to cause confusion. This is also different from a POV from a higher dimension, or perspective. A 4-D "mythical, hypothetical" being, will in fact see all of the 6 sides of a 3-cube, solid or translucent, at the same time. If numbers are written on them, these will show up as well, in a single glance.

Using a 2-D array of photoreceptors inside our eyes, we see 2-D snapshots as 3-D beings. A 2-D being has a 1-D array (line) of photocells, and sees 1-D scans of its 2-D world. With a 4-D being, the eye will have a 3-D array of photoreceptors, which produce an entire 3-D snapshot, capable of seeing all of the insides and outsides of 3-D objects at once. And, again with a 5-D being: using a 4-D array of photocells, a 5-D being can see all insides and outsides of a 4-D object at once, in a single glance. It follows a predictable number sequence. Hope that helps any....

I agree that a 4D-being will be able to see all sides of a cube. However, I think he will have difficulties with looking at what's written on it, because the drawing is 2D, and the creature is looking 3D. (in the same way we have difficulties seeing what's written on the sides of a square). we may see our square as a stack of line segments. there are infinitely many line segments, and only one gives one side. it's pretty hard to see this specific line in between all the other lines, and it's even harder to see if there's something written on it. in the same way, a 4D-creature may look at a cube, and his "picture" can be seen as being build up of planes. Now only one plane is one square of the cube, and thus the creature will have difficulties looking at this as well. These difficulties can be avoided. One way to avoid them is saying the creature is a mathematical thing that can see all points. This mathematical thing won't have difficulties, because it's mathematical. Another way to avoid this is by "prisming" the square/cube. this is a invalid way, because it's not a square/cube anymore. however, this is what I think 4D-space is using, and if you look at it this way, the square/cube-prism will have sides that you indeed won't be able to see all-at-once.

Note that I do NOT agree with 4D-space about quickfur being whatever-he-says. I respect quickfur (and other members of this forum (including 4D-space)), and do not like 4D-space's (and other’s) impolite writings.

How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.-Stern/Multatuli/Eduard Douwes Dekker

Well, I suppose that a written language in 2-D land would be akin to Morse code, a sequence of lines and dots. This being the case, the 1-D ink would have to be superimposed into the 1-D paper, where sections of this paper are missing, to make room for the ink. In this method of analogy, it would then be possible for a 3-D being to read everything written on the surface ( perimeter ) of a square, in a single glance. Only in this way can a 4-D being view the writing on all surfaces of a cube, translucent or not.

Luckily, having worked in retail as a bicycle mechanic for 14 years, I have learned to bite my tongue when someone, passionate in their views, demands aggressively that they are correct. If it doesn't have anything to do with their safety, I let it go, and consider any help a lost cause. I like to avoid playing the " Who's more correct? " game, as in most cases, it's a subjective point of view. "There are many words, but only one truth" - A Prior of the Ori.